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By Abdullah Alturki Lecturer in Mathematics Kingdom of Saudi Arabia Prince Sattam Bin Abdulaziz University College of Science and Humanities in Hotat Bani Tamim Department of Mathematics 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Examples of Real Numbers Properties of Real Numbers Examples for Properties of Real Numbers Natural numbers are the set of counting numbers which starts from 1. They have only positive value numbers. Natural N numbers are denoted by N ={1,2,3,4,5,…} Whole numbers are the set of numbers that include 0 plus the set of natural numbers. They have one number zero as nether positive nor negative but all other numbers are positive. Whole numbers are denoted by W W ={0,1,2,3,4,5,…} W ={0} U N Integers are the set of whole numbers with negative values. Positive Integers Z+ = { 1 , 2 , 3 , 4 , 5 , … } = N Negative Integers Z− = { -1 , -2 , -3 , -4 , -5 , … } Integers Z = { … , -3 , -2 , -1 , 0 , 1 , 2 , 3 , … } ={0,±1,±2,±3,±4,±5,…} They have a zero and the positive numbers with negative numbers. Z+ U { 0 } U Z− = Z Rational numbers are any numbers that can be expressed in the a form of (fractions) , where a and b are integers, and b ≠ 0. b They are written as an integer divided by another integer and the denominator is not zero and both numbers do not have common factors. Rational numbers can be called fractions. b Any number in the form a such that a,b,c Z , c 0 is rational c 1 number . Example: 1 is rational number. 2 They can always be expressed by using terminating decimals or repeating decimals. Terminating decimals are decimals that contain a finite number of 1 3 1 digits. Examples : 0.5 = , 1.5 = , 0.125 = 2 2 8 Repeating decimals are decimals that contain infinite number of digits. 1 5 Examples : 0.3333…= 0.3 = 3 , 0.050505… = 0.05 = 99 Rational numbers are denoted by Q m Q={ n : mZ , nZ , n0 } 5 m Every integer is a rational number. Example 5 = 1 in general m Z , m = 1 Irrational numbers are any numbers that cannot be written as a simple fraction . They are expressed as non-terminating(non-ending),non-repeating decimal. A prime number (or a prime) is a natural number greater than 1 that can be divided, without a remainder, only by 1 and itself. The roots of prime number are irrational = If p is prime number , then 𝑝 is irrational number . Irrational numbers are denoted by Examples for Irrational numbers : Pi = 𝜋 3.14… , e 2.71… , 2 , 3 , - 5 , 1 2 , 2 5 , 1 𝜋 2 The real numbers include all the rational numbers and all the irrational numbers . Real numbers are denoted byR R={ N I x x is rational or x is irrational } WZQ R R Which of the following is a real number ? A. - 21 B. 0.123 C. 3 2 D. E. 13 All of the answer choices are correct. Real Numbers include: A. Natural Numbers and Whole numbers B. Integers C. Rational Numbers and Irrational Numbers D. All of the answer choices are correct. Which classification describes the number 2 A. Rational number B. Irrational number C. Real number D. B and C Which A. i B. 3+i C. −1 D. 4 one of the following is a real number ? Which A. 5 2 B. -5 C. D. 22 7 𝜋 one of the following is not a rational number? Which one of the following is not an irrational number? B. 1 3 e C. 7 A. D. 3 2 The number 0.1212212221222212… is A. Natural Number B. Integer C. Rational Number D. Irrational Number Z+ U { 0 } U Z− = A. Natural Numbers N B. Whole Numbers W C. Integers Z D. Irrational Number I Which one of the following is a real number ? A. i ( Imaginary Number ) B. 3 + −1 C. 2 - −1 D. ( −1 )2 Which A. i B. 3+i C. −1 D. 𝑖2 one of the following is a real number ? Which classification describes the number -2.3 A. Positive number B. Whole number C. Irrational number D. Rational number Is A. B. 0.50 a rational number ? Yes No Is A. B. ( - 0.252525 ) a rational number? Yes No Is A. B. 0.123456789 a rational number? Yes No 20 a rational number ? A. Yes B. No 1 Is ( - 2 ) a rational number ? 5 A. Yes B. No 8 Is a rational number ? 2 A. Yes B. No Is Is A. B. 1 5 a rational number ? Yes No Is 3 5 a rational number ? Yes B. No Is 𝜋 a rational number ? A. Yes B. No A. Consider the following set of numbers. 9 1 { 0 , 1 , -2 , , 16 , 0.1 , 4 , 0.5 , 11 } 3 2 List the numbers in the set that are : Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real numbers List the numbers in the set below that belong to the set of rational numbers. 6 { - , - 9 , - 0.25 , 𝜋 , e , 7 } 3 A. { 𝜋 , e , 7 } 6 B. { - , - 9 , - 0.25 } 3 6 C. { - , - 9 , - 0.25 , e } 3 6 D. { - , - 9 , - 0.25 , 𝜋 } 3 Classifying Real Numbers : Name the set(s) of numbers to which the number belongs to . 0.5 Real number and rational number . 3 Real number and irrational number . 3.14 Real number and rational number . -7 Real number, rational number and integer . Counter Example: an example that proves a statement false . True or False . If false give a counter example . All A. B. whole numbers are rational numbers . True False All A. B. whole numbers are natural numbers . True False 1. Commutative Property of Addition For any real numbers a and b . a+b=b+a 2. Commutative Property of Multiplication For any real numbers a and b . a•b=b•a 3. Associative Property of Addition For any real numbers a , b and c . a+(b+c)=(a+b)+c or (a+b)+c=a+(b+c) 4. Associative Property of Multiplication For any real numbers a , b and c . a•(b•c)=(a•b)•c or (a•b)•c=a•(b•c) 5. Distributive Property For any real numbers a , b and c . a•(b+c)=a•b+a•c or (a+b)•c=a•c+b•c 6. Additive Identity Property For any real numbers a . a+0=a=0+a 7. Multiplicative Identity Property For any real numbers a . a•1=a=1•a 8. Additive Inverse Property For any real numbers a . a+(-a)=0=(-a)+a 9. Multiplicative Inverse Property For any real numbers a . a • ( a𝟏 ) = 1 = ( a𝟏 ) • a , a 0 10. Zero Property For any real numbers a . a•0=0=0•a 11. Closure Property of Addition For any real numbers a and b . a + b is real number . 12. Closure Property of Multiplication For any real numbers a and b . a • b is real number . To combine real numbers involving negatives, we use these properties. x + 3 = 3 + x is an example of which property? A. associative property of addition B. additive identity C. commutative property of addition D. additive inverse 5( a + 1 ) = 5a + 5 is an example of which property? A. associative property of multiplication B. distributive property C. commutative property of multiplication D. multiplicative inverse property a + ( b + 2 ) = a + ( 2 + b ) is an example of which property? A. associative property of addition B. distributive property C. additive identity D. commutative property of addition ( 3 b ) • ( 1 ) = 3 b is an example of which property? A. multiplicative identity property B. multiplicative inverse property C. commutative property of multiplication D. associative property of multiplication ( a b ) c = a ( b c ) is an example of which property? A. commutative property of multiplication B. associative property of multiplication C. distributive property D. multiplicative inverse property ( a + 2 ) ( 3 + a ) = ( a + 2 )•( 3 )+( a + 2 )•( a ) is an example of which property? A. associative property of addition B. commutative property of multiplication C. associative property of multiplication D. distributive property (x + 3) + (-x + -3) = 0 is an example of which property? A. multiplicative identity property B. additive identity property C. multiplicative inverse property D. additive inverse property 1 3y . ( 3𝑦 ) = 1 is an example of which property? A. multiplicative identity property B. multiplicative inverse property C. commutative property of multiplication D. associative property of multiplication Determine which equation illustrates the commutative property. A. x+0=x B. x + (y + z ) = (x + y) + z C. x+y=y+x D. x(y + z) = xy + xz Which property is illustrated by the equation xa + xb = x(a + b) A. Associative B. Commutative C. Distributive D. Identity Decide which property goes along with each of the following examples ( 1 ) = 1438 is … multiplicative identity property 1438 2016 + 0 = - 2016 is … additive identity property - + 50 = 50 + 25 is … commutative property of addition 25 - 5 ) 3 = 3 ( - 5 ) is … commutative property of multiplication ( Decide which property goes along with each of the following examples 3 + ( 2 + 1 ) = ( 3 + 2 ) + 1 is … associative property of addition 4 ( 3 2 ) = ( 4 3 ) 2 is … associative property of multiplication - 9 + 9 = 0 is … additive inverse property 1 - 15 ( - 15 ) = 1 is … multiplicative inverse property Decide which property goes along with each of the following examples 5 ( 3 + 2 ) = 5 3 + 5 2 is … distributive property If x,y Z then x + y Z is … closure property of addition If 2 Z , 5 Z then ( 2 )( 5 ) Z is … closure property of multiplication 3 x + 2 x = ( 3 + 2 ) x is … distributive property The property that justify 5 ( 15 ) = 1 is called A. multiplicative identity property B. multiplicative inverse property C. commutative property of multiplication D. associative property of multiplication The 1 A. 11 B. 1 - 11 C. 11 D. -11 additive inverse of the number 11 is The A. 1 11 B. 1 - 11 C. 11 D. -11 multiplicative inverse of the number 11 is The A. 2 B. - 12 C. -2 D. 1 2 additive inverse of the number - 1 2 is The A. 2 B. - 12 C. -2 D. 1 2 multiplicative inverse of the number - 12 is The additive inverse of the number - 0.1 is A. - 0.1 B. - 10 C. 10 D. 0.1 What A. - 0.1 B. - 10 C. 10 D. 0.1 is multiplicative inverse of - 0.1 ? Use the distributive property to simplify : 5 ( 2x - 3 ) - 2 = … A. 10 x + 17 B. 10 x - 6 C. 7 x - 17 D. 10 x - 17 Use properties of real numbers to simplify algebraic expression : 3x+2y+4x-y Use properties of real numbers to simplify algebraic expression : -3a-b-4a-5b Use properties of real numbers to simplify algebraic expression : 2(3x+y)+5(4x-y) Use properties of real numbers to simplify algebraic expression : 4 ( 3 x - y ) - 2 ( 5x + 4y ) Use properties of real numbers to simplify algebraic expression : 1 1 -2(8x-2y)-3(6x-3y) Use properties of real numbers to simplify algebraic expression : 5 𝑦 − 2 − 5𝑦 + 3 − 3 3𝑦 − 2 Use properties of real numbers to simplify algebraic expression : 𝑥− 𝑥− 𝑥− 𝑥−1 What is the identity element for addition of real numbers? A. -1 B. 1 −1 C. D. 0 What is the identity element for multiplication of real numbers? A. -1 B. 1 −1 C. D. 0 A set S is said to be closed under addition if for all elements a and b in S, a + b is also in S. Which one of the following sets is closed under addition? A. {0} B. {1} C. {0,1} A set S is said to be closed under multiplication if for all elements a and b in S, a × b is also in S. Which of the following sets is not closed under multiplication? A. {0} B. {1} C. { 1 , -1 } D. {1,2} Closure: When you combine any two elements of the set, the result is also included in the set. If you add two even numbers (from the set of even numbers), is the sum even? A. B. Yes No Closure: When you combine any two elements of the set, the result is also included in the set. If you add two natural numbers (from the set of natural numbers), is the sum natural number ? A. B. Yes No A. B. If you divide two even numbers (from the set of even numbers), is the quotient (the answer) even? Yes No